Expressions, Equations and Inequalities: Grade 7 Study Guide
Equations model real situations, and solving them tells us what we need to know.
To simplify an expression, combine like terms and use the distributive property: 3(2x − 5) + 4x = 6x − 15 + 4x = 10x − 15. Factoring does the reverse: 12x + 18 = 6(2x + 3). Different but equivalent expressions can make a problem easier to see, such as writing 'a 15% increase' as 1.15 times the original.
A two-step equation such as 3x + 5 = 20 needs two inverse operations: subtract 5 (3x = 15), then divide by 3 (x = 5). An inequality like 3x + 5 > 20 has many solutions (x > 5), shown on a number line.
Solving two-step equations
3x + 5 = 20 → 3x = 15.
x = 5.
3(5) + 5 = 20.
x = 5 tickets.
Inequalities
Solve like an equation, but reverse the inequality sign when multiplying or dividing by a negative number. −2x < 10 → x > −5. Graph with an open circle for < or >, and a closed circle for ≤ or ≥. Real-life example: you have $50 and a movie costs $8.50 per ticket plus a $3 fee. 8.5t + 3 ≤ 50 → t ≤ 5.52, so at most 5 tickets.
| Inequality | Meaning | Graph |
|---|---|---|
| x > 3 | greater than 3 | open circle, arrow right |
| x ≤ 7 | at most 7 | closed circle, arrow left |
| −2 < x < 4 | between | open circles at both ends |
Word problems
Define a variable, write an equation, solve and check that the answer makes sense. A plumber charges $45 plus $30 per hour. If the bill is $150: 30h + 45 = 150 → 30h = 105 → h = 3.5 hours. A square with a perimeter of 36 cm has sides of 9 cm: 4s = 36.
Worked examples and common mistakes
Example: Solve 2(x − 3) = 14: x − 3 = 7, x = 10. Common mistakes: forgetting to distribute to every term, not flipping the inequality sign for a negative divisor and answering with a number that does not fit the situation (such as 5.52 tickets). Always interpret the solution.
Check your understanding: model answers
1. x = 7. 2. x ≤ −4. 3. 5x + 10 − 3x = 2x + 10.
✨ Key Takeaways
- Combine like terms and distribute.
- Use inverse operations in reverse order.
- Flip the inequality sign when multiplying or dividing by a negative.
- Interpret the answer in context.
💬 Questions for Discussion & Reflection
- Solve 4x − 7 = 21.
- Solve −3x ≥ 12.
- Simplify 5(x + 2) − 3x.